{
  "id": "1410.6320",
  "version": "v1",
  "published": "2014-10-23T11:21:29.000Z",
  "updated": "2014-10-23T11:21:29.000Z",
  "title": "Copies of the Random Graph",
  "authors": [
    "Miloš S. Kurilić",
    "Stevo Todorčević"
  ],
  "comment": "28 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $(R, \\sim )$ be the Rado graph, $Emb (R)$ the monoid of its self-embeddings, $\\Pi (R)=\\{ f[R]: f\\in Emb (R)\\}$ the set of copies of $R$ contained in $R$, and ${\\mathcal I}_R$ the ideal of subsets of $R$ which do not contain a copy of $R$. We consider the poset $( \\Pi (R ), \\subset )$, the algebra $P (R)/{\\mathcal I _R}$, and the inverse of the right Green's pre-order on $Emb (R)$, and show that these pre-orders are forcing equivalent to a two step iteration of the form $P \\ast \\pi$, where the poset $P$ is similar to the Sacks perfect set forcing: adds a generic real, has the $\\aleph _0$-covering property and, hence, preserves $\\omega _1$, has the Sacks property and does not produce splitting reals, while $\\pi$ codes an $\\omega$-distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy of the Rado graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-23T11:21:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "03C15",
      "03C50",
      "03E40",
      "06A06",
      "20M20"
    ],
    "keywords": [
      "random graph",
      "rado graph",
      "right greens pre-order",
      "boolean completions",
      "sacks perfect set forcing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.6320K"
    }
  }
}